Problem: Determine how many solutions exist for the system of equations. ${-6x+3y = -12}$ ${-5x+y = 7}$
Solution: Convert both equations to slope-intercept form: ${-6x+3y = -12}$ $-6x{+6x} + 3y = -12{+6x}$ $3y = -12+6x$ $y = -4+2x$ ${y = 2x-4}$ ${-5x+y = 7}$ $-5x{+5x} + y = 7{+5x}$ $y = 7+5x$ ${y = 5x+7}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x-4}$ ${y = 5x+7}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.